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Ερμιτιανή Μήτρα
Ερμιτιανή Μήτρα Hermitian Matrix thumb|300px| [[Μοναδιακή Μήτρα Ερμιτιανή Μήτρα ]] thumb|300px| [[Μοναδιακή Μήτρα Ερμιτιανή Μήτρα ]] 300px|thumb| [[Ερμιτιανή Μήτρα ]] thumb|300px| [[Ερμιτιανή Μήτρα ]] ---- Μιγαδική Συζυγής Μήτρα Ανάστροφη Μήτρα Ερμιτιανή Μήτρα]] - Ένα είδος Μήτρας. Ετυμολογία Η ονομασία "Ερμιτιανός" σχετίζεται ετυμολογικά με το όνομα "Hermite" who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real.. Εισαγωγή In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose – that is, the element in the i''-th row and ''j-th column is equal to the complex conjugate of the element in the j''-th row and ''i-th column, for all indices i'' and ''j: : a_{ij} = \overline{a_{ji}}\,. If the conjugate transpose of a matrix A is denoted by A^\dagger , then the Hermitian property can be written concisely as : A = A^\dagger\,. Hermitian matrices can be understood as the complex extension of real symmetric matrices. Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real. Examples For example, : \begin{bmatrix}3&2+i\\ 2-i&1\end{bmatrix}. Well-known families of Pauli matrices, Gell-Mann matrices and various generalizations are Hermitian. In theoretical physics such Hermitian matrices usually are multiplied by imaginary coefficients, Physics 125 Course Notes at California Institute of Technology which results in skew-Hermitian matrices (see below). Properties The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real. A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. (A real and symmetric matrix is simply a special case of a Hermitian matrix). Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies. It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix A'' are real, and that ''A has n'' linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of '''Cn'' consisting of n'' eigenvectors of ''A. *The sum of any two Hermitian matrices is Hermitian, and *the inverse of an invertible Hermitian matrix is Hermitian as well. *However, the product of two Hermitian matrices A'' and ''B is Hermitian if they commute, i.e., if AB = BA. Thus ''A'n'' is Hermitian if A'' is Hermitian and ''n is an integer. The Hermitian complex n''-by-''n matrices do not form a vector space over the complex numbers, since the identity matrix I_n is Hermitian, but i(I_n) is not. However the complex Hermitian matrices do form a vector space over the real numbers. In the 2n''2 '''R' dimensional vector space of complex n''×''n matrices, the complex Hermitian matrices form a subspace of dimension n''2. If ''E''jk denotes the ''n-by-''n'' matrix with a 1 in the j'',''k position and zeros elsewhere, a basis can be described as follows: : \; E_{jj} for 1\leq j\leq n (n'' matrices) together with the set of matrices of the form : \; E_{jk}+E_{kj} for 1\leq j ((''n''2−''n)/2 matrices) and the matrices : \; i(E_{jk}-E_{kj}) for 1\leq j ((n''2−''n)/2 matrices) where i denotes the complex number \sqrt{-1} , known as the imaginary unit. If n'' orthonormal eigenvectors u_1,\dots,u_n of a Hermitian matrix are chosen and written as the columns of the matrix ''U, then one eigendecomposition of A'' is A = U \Lambda U^\dagger where U U^\dagger = I=U^\dagger U and therefore : A = \sum _j \lambda_j u_j u_j ^\dagger , where \lambda_j are the eigenvalues on the diagonal of the diagonal matrix \; \Lambda . Additional facts related to Hermitian matrices include: * The sum of a square matrix and its conjugate transpose (C + C^{\dagger}) is Hermitian. * The difference of a square matrix and its conjugate transpose (C - C^{\dagger}) is skew-Hermitian (also called antihermitian). ** This implies that commutator of two Hermitian matrices is skew-Hermitian. * An arbitrary square matrix ''C can be written as the sum of a Hermitian matrix A'' and a skew-Hermitian matrix ''B: :: C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^{\dagger}) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^{\dagger}). * The determinant of a Hermitian matrix is real: :: Proof: \det(A) = \det(A^\mathrm{T})\quad \Rightarrow \quad \det(A^\dagger) = \det(A)^* :: Therefore if A=A^\dagger\quad \Rightarrow \quad \det(A) = \det(A)^*. :(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.) Pauli spin matrices Any 2x2 Hermitian matrix may be written as a linear combination of the 2×2 identity matrix and the three Pauli spin matrices. These matrices have use in quantum mechanics. The four matrices form an orthogonal basis for the 4-dimensional vector space of 2x2 Hermitian matrices. : \mathbf{I}=\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \sigma_{x}=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix},\quad \sigma_{y}=\begin{pmatrix}0 & -\mathit{i} \\ \mathit{i} & 0 \end{pmatrix},\quad \sigma_{z}=\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} An arbitrary 2×2 Hermitian matrix A is written thus, : \mathbf{A} = a\mathbf{I} + b\sigma_{x} + c\sigma_{y} + d\sigma_{z} : A = \begin{pmatrix}a & 0 \\ 0 & a \end{pmatrix} + \begin{pmatrix}0 & b \\ b & 0 \end{pmatrix} + \begin{pmatrix}0 & -\mathit{i}c \\ \mathit{i}c & 0 \end{pmatrix} + \begin{pmatrix}d & 0 \\ 0 & -d \end{pmatrix} : A =\begin{pmatrix}a+d & b-\mathit{i}c \\ b+\mathit{i}c & a-d \end{pmatrix} Skew-Hermitian Matrices A skew-Hermitian matrix is one which is equal to the negative of its Hermitian adjoint: : \mathbf{A=-A^*} For instance, (a'', ''b, c'', ''d, e'', ''f, g'', ''h, and k are real), : \mathbf{A}=\begin{pmatrix} \mathit{i}a & -b+\mathit{i}c & -e+\mathit{i}f \\ b+\mathit{i}c & \mathit{i}d & -h+\mathit{i}k \\ e+\mathit{i}f & h+\mathit{i}k & \mathit{i}g \end{pmatrix} is a skew-Hermitian matrix. Clearly, the entries on the main diagonal are purely imaginary. Εφαρμογή Σε κάθε Φυσικό Μέγεθος αντιστοιχεί ένας κατάλληλος Ερμιτιανός Τελεστής που αναπαρίσταται από μία Ερμιτιανή Μήτρα, του οποίου οι ιδιοτιμές είναι τα μοναδικά δυνατά εξαγόμενα μιας μέτρησης. Υποσημειώσεις Εσωτερική Αρθρογραφία *τελεστής *Μαθηματική Μήτρα *Ορθογώνια Μήτρα *Μοναδιακή Μήτρα *Hermitian form *Self-adjoint operator *Haynsworth inertia additivity formula Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation. *[ ] Category: Μήτρες